Flux-Tunable Qubit Device with Multiple Josephson Junctions

ABSTRACT

In a general aspect, a qubit device includes two circuit loops. In some aspects, a first circuit loop includes a first Josephson junction, a second circuit loop includes a second Josephson junction, and the first and second loops are configured to receive a magnetic flux that defines a transition frequency of a qubit device. In some aspects, a quantum integrated circuit includes an inductor connected between a first circuit node and a second circuit node; the first Josephson junction connected in parallel with the inductor between the first circuit node and the second circuit node; and the second Josephson junction connected in parallel with the inductor between the first circuit node and the second circuit node.

BACKGROUND

The following description relates to a flux-tunable qubit device thatincludes multiple Josephson junctions.

In some quantum computing architectures, qubits are implemented insuperconducting circuits. The qubits can be implemented, for example, incircuit devices that include Josephson junctions. Examples includetransmon qubits, flux qubits and fluxonium qubits.

DESCRIPTION OF DRAWINGS

FIG. 1 is a bock diagram of an example quantum computing system.

FIG. 2A shows an equivalent circuit for an example qubit device.

FIG. 2B shows an equivalent circuit for another example qubit device.

FIG. 2C shows an equivalent circuit for an example quantum integratedcircuit.

FIGS. 3A and 3B show an example quantum integrated circuit.

FIG. 4 is a plot showing transition frequencies as a function ofexternal magnetic flux for an example qubit device.

FIGS. 5A and 5B are plots showing transition frequencies as a functionof external magnetic flux for example qubit devices.

FIG. 6 is a plot showing the derivative of a transition frequency as afunction of external magnetic flux for an example qubit device.

DETAILED DESCRIPTION

In some aspects of what is described here, a quantum circuit includesmultiple tunable circuit elements. In some implementations, the tunablecircuit elements provide in-situ tunability in the quantum circuit,which can be an advantageous resource, for instance, for constructingand operating quantum computers and for other applications. For example,incorporating tunable quantum circuits in a quantum processor may reduceprecision requirements for fabrication of the quantum processor. Asanother example, some quantum algorithms may be implemented on a quantumprocessor by tuning the quantum circuits to perform operations (e.g.,quantum gates or other operations). In some implementations, a tunablequantum circuit can be designed to reduce sensitivity to a dominantnoise source, and thereby reduce errors and improve performance of thequantum processor.

Superconducting Quantum Interference Devices (SQUIDs) are an example ofa tunable circuit element that can be implemented in a superconductingquantum integrated circuit, in some cases. A SQUID typically includes asuperconducting loop with one or more Josephson junctions embedded. Theresonance frequency of a SQUID loop depends on the magnetic flux throughthe superconducting loop. SQUIDs can be used to createmagnetic-flux-tunable circuits, including qubit devices having magneticflux-tunable qubit frequencies.

FIG. 1 is a schematic diagram of an example quantum computing system100. The example quantum computing system 100 shown in FIG. 1 includes acontrol system 110, a signal delivery system 106, and a quantum circuitsystem 102. A quantum computing system may include additional ordifferent features, and the components of a quantum computing system mayoperate as described with respect to FIG. 1 or in another manner.

The example quantum computing system 100 shown in FIG. 1 can performquantum computational tasks by executing quantum algorithms. In someimplementations, the quantum computing system 100 can perform quantumcomputation by storing and manipulating information within individualquantum states of a composite quantum system. For example, qubits (i.e.,quantum bits) can be stored in and represented by an effective two-levelsub-manifold of a quantum coherent physical system. Coupler devices canbe used to perform quantum logic operations on single qubits orconditional quantum logic operations on multiple qubits. In someinstances, the conditional quantum logic can be performed in a mannerthat allows large-scale entanglement within the quantum computingdevice. Control signals can manipulate the quantum states of individualqubits and the joint states of multiple qubits. In some instances,information can be read out from the composite quantum system bymeasuring the quantum states of the qubit devices.

The example quantum circuit system 102 shown in FIG. 1 is asuperconducting quantum integrated circuit that includes qubit devices.The qubit devices are used to store and process quantum information, forexample, by operating as ancilla qubits, data qubits or other types ofqubits in a quantum algorithm. In some instances, all or part of thequantum circuit system 102 functions as a quantum processor, a quantummemory, or another type of subsystem. The quantum circuit system 102shown in FIG. 1 can include qubit devices or other devices that areimplemented according to the examples shown in FIG. 2A, 2B, 2C, 3A or3B, or in another manner.

In some implementations, the quantum circuit system 102 includes atwo-dimensional or three-dimensional device array, which includesdevices arranged in a lattice structure. For instance, a two-dimensionaldevice array can be formed on a two-dimensional wafer surface, where thedevices (e.g., qubit devices) are arranged in a two-dimensional latticestructure and configured to communicate with one another. Athree-dimensional device array can be formed by a stack oftwo-dimensional wafers, where the devices are arranged in athree-dimensional lattice structure and configured (e.g., by connectionsbetween wafers) to communicate with one another. In someimplementations, an electromagnetic waveguide system provides anenvironment for the device array. For instance, some or all of thequantum circuit system 102 can be housed in an electromagnetic waveguidesystem that provides a low-noise electromagnetic environment for thequbit devices.

The example quantum circuit system 102, and in some cases all or part ofthe signal delivery system 106, can be maintained in a controlledcryogenic environment. The environment can be provided, for example, byshielding equipment, cryogenic equipment, and other types ofenvironmental control systems. In some examples, the components in thequantum circuit system 102 operate in a cryogenic temperature regime andare subject to very low electromagnetic and thermal noise. For example,magnetic shielding can be used to shield the system components fromstray magnetic fields, optical shielding can be used to shield thesystem components from optical noise, thermal shielding and cryogenicequipment can be used to maintain the system components at controlledtemperature, etc.

In the example quantum circuit system 102, the qubit devices each storea single qubit (a bit of quantum information), and the qubits cancollectively define the computational state of a quantum processor orquantum memory. In some implementations, qubit devices in the quantumcircuit system 102 can each be encoded with a single bit of quantuminformation. For instance, each of the qubit devices can define twoeigenstates that are used as computational basis states (“0” and “1”),and each qubit device can transition between its computational basisstates or exist in an arbitrary superposition of its basis states.

The example quantum circuit system 102 may also include readout devicesthat selectively interact with the qubit devices to detect their quantumstates. For example, the readout devices may provide readout responsesignals that indicate the computational state of the quantum processoror quantum memory. The readout resonator 284 shown in FIG. 2C and thereadout resonator 304 shown in FIG. 3A are examples of readout devices.The quantum circuit system 102 may also include coupler devices thatselectively operate on individual qubits or pairs of qubits. Forexample, the coupler devices may produce entanglement or othermulti-qubit states over two or more qubits in the quantum circuit system102.

In some examples, individual devices (e.g., qubit devices, couplerdevices, etc.) in the quantum circuit system 102 each have a tunableoperating frequency. For example, the operating frequency can be tunedby applying an offset field. As a particular example, a device mayinclude one or more circuit loops (e.g., SQUID loops or other types ofcircuit loops) configured to receive a magnetic flux that defines anoperating frequency of the device, and the operating frequency may bechanged (increased or decreased) by controlling a magnetic field (e.g.,an external magnetic field that provides the magnetic flux).

In some implementations, qubit devices in the quantum circuit system 102each include a first circuit loop and a second circuit loop configuredto receive a magnetic flux that defines the qubit frequency of the qubitdevice. The circuit loops can be configured such that the qubitfrequency as a function of the magnetic flux has one or more flux sweetspots. For example, the flux sweet spots shown in FIGS. 4 and 5 are theflux points at which the derivatives (first order and higher orderderivatives) of the qubit frequency are substantially zero. In somecases, the qubit frequency has one or more flux sweet spots between themaximum and minimum qubit frequencies over one or more flux quanta. Insome instances, tuning a qubit device to a flux sweet spot can improveoperation of the qubit device, for example, by reducing sensitivity tomagnetic flux noise. In some implementations, the magnetic flux areas ofthe circuit loops, the external magnetic field applied to the circuitloops, the Josephson energies of the Josephson junctions in the circuitloops or a combination of these and other parameters can be selectedsuch that multiple flux sweet spots are available.

In some instances, operating a qubit device at a flux sweet spot canreduce or eliminate errors (e.g., qubit dephasing) produced bystochastic fluctuations in the external magnetic field. In somecontexts, qubit dephasing can be understood in terms of fluctuations ofthe qubit transition frequency due to its coupling to the environment.This can be described as noise in the external flux parameters (Φ_(ext))of the Hamiltonian. The fluctuation generally results in two distincteffects: (1) for sufficiently low frequencies, the fluctuations cancause random shifts of the transition frequency of the qubit, leading topure dephasing, on the time scale of T₂; (2) for higher frequencies, thefluctuations can induce transition between qubit states, leading toenergy relaxation, on the time scale of Engineering flux sweet spots canreduce the dephasing rate, thus improving the characteristic dephasingtime T₂. Such improvements can increase quantum gate fidelity orotherwise improve operation of the qubit device.

In some implementations, the example quantum circuit system 102 canprocess the quantum information stored in the qubits by applying controlsignals to the qubit devices or other devices (e.g., to coupler devices)housed in a quantum processor cell. The control signals can beconfigured to encode information in the qubit devices, to process theinformation by applying logical gates or other types of operations, orto extract information from the qubit devices. In some examples, theoperations can be expressed as single-qubit gates, two-qubit gates, orother types of logical gates that operate on one or more qubits. Asequence of operations can be applied to the qubits to perform a quantumalgorithm. The quantum algorithm may correspond to a computational task,a quantum error correction procedure, a quantum state distillationprocedure, or a combination of these and other types of operations.

In the example shown in FIG. 1, the signal delivery system 106 providescommunication between the control system 110 and the quantum circuitsystem 102. For example, the signal delivery system 106 can receivecontrol signals from the control system 110 and deliver the controlsignals to the quantum circuit system 102. In some instances, the signaldelivery system 106 performs preprocessing, signal conditioning, orother operations to the control signals before delivering them to thequantum circuit system 102.

In some implementations, the signal delivery system 106 includesconnectors or other hardware elements that transfer signals between thequantum circuit system 102 and the control system 110. For example, theconnection hardware can include signal lines, signal processinghardware, filters, feedthrough devices (e.g., light-tight feedthroughs,etc.), and other types of components. In some implementations, theconnection hardware can span multiple different temperature and noiseregimes. For example, the connection hardware can include a series oftemperature stages (60 K, 3 K, 800 mK, 150 mK) that decrease between ahigher temperature regime (e.g., at the control system 110) and a lowertemperature regime (e.g., at the quantum circuit system 102).

In the example quantum computing system 100 shown in FIG. 1, the controlsystem 110 controls operation of the quantum circuit system 102. Theexample control system 110 may include data processors, signalgenerators, interface components and other types of systems orsubsystems. In some cases, the control system 110 includes one or moreclassical computers or classical computing components. Components of theexample control system 110 may operate in a room temperature regime, anintermediate temperature regime, or both. For example, the controlsystem 110 can be configured to operate at much higher temperatures andbe subject to much higher levels of noise than are present in theenvironment of the quantum circuit system 102.

In some implementations, the control system 110 includes a classicalcomputing system that executes software to compile instructions for thequantum circuit system 102. For example, the control system 110 maydecompose a quantum logic circuit into discrete control operations orsets of control operations that can be executed by the hardware in thequantum circuit system 102. In some examples, that control system 110applies a quantum logic circuit by generating signals that cause thequbit devices and other devices in the quantum circuit system 102 toexecute operations. For instance, the operations may correspond tosingle-qubit gates, two-qubit gates, qubit measurements, etc. Thecontrol system 110 can generate control signals that are communicated tothe quantum circuit system 102 by the signal delivery system 106, andthe devices in the quantum circuit system 102 execute the operations inresponse to the control signals.

In some aspects of operation, information is encoded in data qubitsresiding in the quantum circuit system 102. For example, a single qubitof information may be written to, stored on or otherwise encoded in eachdata qubit. In some cases, to encode information in the data qubits, thecontrol system 110 sends control signals to the quantum circuit system102. The control signals can each be addressed to individual qubitdevices, and can be configured to modify the quantum states of therespective qubits to which they are addressed. For example, the controlsignals may be configured to transit the qubit devices to a particularcomputational state, to apply logical operations to the qubits, or tootherwise encode information in the qubit devices.

In some aspects of operation, the control system 110 includes amicrowave signal source (e.g., an arbitrary waveform generator), a biassignal source (e.g., a direct current source) and other components thatprovide control signals to the signal delivery system 106. The controlsignals can include analog signals that are generated based on digitalcontrol information provided, for instance, by a classical processor inthe control system 110. In some instances, the control signals aregenerated by the control system 110 at high temperature (e.g., abovecryogenic temperatures) and delivered to the quantum circuit system 102operating at low temperature (e.g., at cryogenic temperatures). Withinthe quantum circuit system 102, the control signals can be delivered toindividual circuit devices, for instance, to apply quantum logic gates,readout qubit states or to perform other operations.

In some aspects of operation, the quantum circuit system 102 producessignals that are delivered to the control system 110 by the by thesignal delivery system 106. For example, readout devices may producereadout response signals (e.g., in response to a readout interrogationsignal) that are transferred to the signal delivery system 106. Thereadout response signals can include analog signals that are produced atlow temperature and delivered to the control system 110 operating at ahigher temperature. The control system 110 may include conversionhardware that digitizes the readout response signals to be processed,for example, by a classical processor in the control system 110.

In some aspects of operation, the control system 110 operates qubitdevices in the quantum circuit system 102. For example, operating aqubit device may include controlling a magnetic flux received by one ormore circuit loops (e.g., SQUID loops) of the qubit device, andcontrolling the magnetic flux can tune the transition frequencies of thequbit device. In some instances, controlling the magnetic flux caninclude tuning the qubit frequency of the qubit device to a flux sweetspot, and the control system 110 can perform a readout of the qubitdevice while the transition frequency is parked at the flux sweet spot.For example, the control system 110 may control the magnetic flux bysending a DC bias signal to a flux bias device that is coupled to thequbit device, and the control system 110 may perform a readout bysending an microwave readout interrogation signal to a readout devicethat is coupled to the qubit device. In some instances, controlling themagnetic flux can include tuning the transition frequency of the qubitdevice away from a flux sweet spot, and the control system 110 can applya logic gate to the qubit device while the transition frequency isparked away from the flux sweet spot. For example, the control system110 may control the magnetic flux by sending a DC bias signal to a fluxbias device that is coupled to the qubit device, and the control system110 may apply a logic gate by sending a microwave control signal to thequbit device.

FIG. 2A is a circuit diagram showing an equivalent circuit for anexample qubit device 200. The example qubit device 200 represented inFIG. 2A includes a first Josephson junction (J₁) 202A, a secondJosephson junction (J₂) 202B, an inductor 204, and a capacitor 208. Thefirst Josephson junction (J₁) 202A has a first inductance (L_(J) ₁ ) anda first Josephson energy (E_(J) ₁ ); the second Josephson junction (J₂)202B has a second inductance (L_(J) ₂ ) and a second Josephson energy(E_(J) ₂ ); the inductor 204 has an inductance L_(A); and the capacitor208 has a capacitance C_(J). In the example shown in FIG. 2A, the firstand second Josephson junctions each include a respective parasiticcapacitance, which is represented by the capacitance C_(J) of thecapacitor 208. The example qubit device 200 may include additional ordifferent features, including connections with other circuit devices, insome cases.

In some cases, the inductor 204 can be implemented as a superinductor. Asuperinductor can be implemented, for example, as a serial array ofJosephson junctions (e.g., as shown in FIG. 3B or otherwise), as asuperconducting wire (e.g., a superconducting wire having a high kineticinductance) or as another type of structure. In some cases, thecapacitor 208 can be implemented as a physical capacitor device in thequantum integrated circuit. In some cases, the capacitance C_(J) of thecapacitor 208 represents the parasitic capacitance of the first andsecond Josephson junctions.

In the example qubit device 200 represented in FIG. 2A, the inductor 204is connected between a first circuit node 212A and a second circuit node212B; the first Josephson junction (J₁) 202A is connected in parallelwith the inductor 204 between the first circuit node 212A and the secondcircuit node 212B; the second Josephson junction (J₂) 202B is connectedin parallel with the inductor 204 between the first circuit node 212Aand the second circuit node 212B. As shown, the parasitic capacitanceC_(J) of the Josephson junctions, represented by the capacitor 208, isconnected in parallel with the inductor 204. The circuit elements may beconnected in another manner in some cases.

In some implementations, the example qubit device 200 shown in FIG. 2Acan be implemented as a superconducting quantum integrated circuit. Forexample, the qubit device 200 may be fabricated on a substrate (e.g., asapphire substrate, a silicon substrate, etc.) that supports one or morelayers of superconducting material (e.g., aluminum, niobium, etc.) andinsulating materials (e.g., aluminum oxide). For example, the Josephsonjunctions 202A, 202B may be formed by a layer of aluminum oxidesandwiched between layers of aluminum superconducting material. In somecases, the inductor 204 can be formed by a series of such Josephsonjunctions. Connections between devices may be formed by superconductingwire (e.g., aluminum).

As shown in FIG. 2A, the example qubit device 200 includes three circuitloops—a first circuit loop 210A, a second circuit loop 210B and a thirdcircuit loop 210C. The first circuit loop 210A includes the firstJosephson junction (J₁) 202A and the second Josephson junction (J₂)202B; the second circuit loop 210B includes the second Josephsonjunction (J₂) 202B and the inductor 204; and the third circuit loop 210Cincludes the first Josephson junction (J₁) 202A and the capacitor 208.The qubit device 200 may include additional or different circuit loopsin some cases. In some instances, the qubit device 200 does not includethe third circuit loop 210C, for example, where the capacitor 208represents the parasitic capacitance of the Josephson junctions.

In some example quantum integrated circuits, a circuit loop can beimplemented as a continuous loop defining an interior area that canreceive magnetic flux during operation of the quantum integratedcircuit. For example, the interior area of the circuit loop can bedefined by the boundaries of circuit elements (e.g., conductors,devices, etc.) on one or more chips or wafers. In some examples, acircuit loop includes circuit elements that define an interior perimeterof the circuit loop, and the interior perimeter defines the boundary ofthe magnetic flux area. In some cases, the magnetic flux (along withelectronic attributes of the circuit elements) defines an operatingfrequency or other operating parameters of the circuit loop or a devicethat includes the circuit loop.

In the example shown in FIG. 2A, the first and second circuit loops210A, 210B are configured to receive, during operation of the qubitdevice 200, a magnetic flux that defines a transition frequency of thequbit device 200. As shown, the first circuit loop 210A has a firstinterior area (A₁) configured to receive a first portion Φ_(ext,1) ofthe external magnetic flux; and the second circuit loop 210B has asecond interior area (A₂) configured to receive a second portionΦ_(ext,2) of the external magnetic flux. As shown in FIG. 2A, the firstinterior area (A₁) and the second interior area (A₂) are exclusiveareas. For instance, the first interior area (A₁) does not contain anyportion of the second interior area (A₂), and vice versa.

The external magnetic fluxes (Φ_(ext,1), Φ_(ext,2)) are produced by anexternal magnetic field threading through the interior areas (A₁, A₂).The external magnetic field can be generated by one or more sources,which may include magnetic field sources on the chip with the qubitdevice, magnetic field sources external to the chip or external to thequantum processor, one or more noise sources in the environment or acombination of them. As an example, in some cases, to generate theexternal magnetic field, a DC signal can be applied to an inductor thathas a mutual inductance with the first and second circuit loops 210A,210B.

In some examples, the external magnetic fluxes (Φ_(ext,1), Φ_(ext,2))are produced by global flux biasing. Global flux biasing can beimplemented, for instance, by using a large coil to generate arelatively uniform magnetic field over the spatial extent of the qubitdevice. The coil can be located below or above the chip, about the chip,or in another location that provides inductive coupling with the circuitloops of the qubit device. When the magnetic field over the spatialextent of the magnetic flux areas (A₁, A₂) is uniform, the ratio of theexternal magnetic fluxes equals the ratio of the magnetic flux areas(r_(Φ)=Φ_(ext,1)/Φ_(ext,2)=A₁/A₂ in some instances).

In some examples, the external magnetic fluxes (Φ_(ext,1), Φ_(ext,2))are produced by on-chip flux biasing. On-chip flux biasing can beimplemented, for instance, by using an inductor on the chip substrate(e.g., near the qubit device 200) to generate a magnetic field. On-chipflux biasing can produce a spatially-varying magnetic field over thespatial extent of the qubit device. For instance, the external magneticfield may be radially-dependent with respect to the flux bias device.When the magnetic field over the spatial extent of the magnetic fluxareas (A₁, A₂) is not uniform, the ratio of the external magnetic fluxesmay be unequal to the ratio of the magnetic flux areas(r_(Φ)=Φ_(ext,1)/Φ_(ext,2)≠A₁/A₂ in some instances).

In some instances, transition frequencies of the qubit device 200 can bemodified (increased or decreased) by controlling the external magneticflux, for example, by modifying (increasing or decreasing) the strengthof an external magnetic field applied the qubit device 200. Thetransition frequencies defined by the external magnetic flux can be thefrequencies associated with energy differences between the quantumenergy levels of the qubit device 200, for example, between the groundstate and the first excited state. In some cases, the qubit frequency,which is the transition frequency between the computational basis statesof the qubit device 200, is among the transition frequencies defined bythe external magnetic flux.

In some implementations, the example qubit device 200 can be configuredto provide a flux sweet spot in the qubit frequency (e.g., thetransition frequency between the ground state and first excited state ofthe qubit). For example, values of the Josephson energies and theinterior areas can be selected such that the qubit device 200 hasmultiple flux sweet spots. In some instances, the qubit device 200 isconfigured to have flux sweet spots, for example, by settingΦ_(ext,1)>Φ_(ext,2), such that the external magnetic flux Φ_(ext,1)through the first circuit loop 210A is larger than the external magneticflux Φ_(ext,2) through the second circuit loop 210B.

To analyze the dephasing time (T₂) in the case of weak fluctuations inthe external magnetic field applied to the example qubit device 200shown in FIG. 2A, each external parameter in the qubit Hamiltonian canbe decomposed into its controlled DC value and fluctuations around it,for example,

Φ_(ext,2)=Φ_(ext,2) ⁽⁰⁾+δΦ_(ext,2).

The Hamiltonian of the qubit, expressed in terms of Pauli operators, canbe expanded in Taylor series as

$H_{q} = {{\frac{1}{2}\left\lbrack {{\hslash\omega}_{01} + {\hslash \frac{{\partial\omega_{\; 01}}\;}{\partial\Phi_{{ext},2}}{\delta\Phi}_{{ext},2}} + {\frac{\hslash^{2}}{2}\frac{\partial^{2}\omega_{01}}{\partial\Phi_{{ext},2}^{2}}{\delta\Phi}_{{ext},2}^{2}} + \ldots} \right\rbrack}{\sigma_{z}.}}$

Considering the first two terms in the Hamiltonian, the decay ofsuperposition state or the off-diagonal density matrix elements can beexpressed

${{\rho_{01}(t)} = {e^{i\; \omega_{01}t}{\langle{\exp \left\lbrack {i\; \delta \; {v(t)}} \right\rbrack}\rangle}}},\; {{\delta \; {v(t)}} = {\frac{{\partial\omega_{01}}\;}{\partial\Phi_{{ext},2}}{\int_{0}^{t}\ {{dt}^{\prime}{{\delta\Phi}_{{ext},2}\left( t^{\prime} \right)}}}}},$

where δv(t) is the random noise accumulated. Assuming Gaussian noise,one can calculate the decay law as

${\langle{\exp \left\lbrack {i\; \delta \; {v(t)}} \right\rbrack}\rangle} = {{\exp \left( {- \frac{{\langle{\delta \; v}\rangle}^{2}}{2}} \right)} = {{\exp \left\lbrack {{- \frac{t^{2}}{2}}\left( \frac{\partial\omega_{01}}{\partial\Phi_{{ext},2}} \right)^{2}{\int_{- \infty}^{\infty}{d\; \omega \mspace{14mu} {s_{\Phi}(\omega)}{{\sin c}^{2}\left( \frac{\omega \; t}{2} \right)}}}} \right\rbrack}.}}$

For noise spectra singular at ω=0, a spectral density can beapproximated as

${S_{\Phi_{2}} = \frac{2\pi \; A_{\Phi}^{2}}{\omega^{\mu}}},$

where 0.8<μ<1.3 and A_(Φ) is the amplitude of the noise. For flux noiseA_(Φ), =10⁻⁶Φ₀−10⁻⁵Φ₀. Introducing infrared and ultraviolet cut offfrequencies ω_(i) and ω_(uv), the dephasing becomes

${{\langle{\exp \left\lbrack {i\; \delta \; {v(t)}} \right\rbrack}\rangle} = {\exp \left\lbrack {{- {A^{2}\left( \frac{\partial\omega_{01}}{\partial\Phi_{{ext},2}} \right)}^{2}}t^{2}{{\ln \; \omega_{i}t}}} \right\rbrack}}\;$

which is valid for ω_(i)t<<1 and ω_(u)t>>1. The dephasing rate due tothe flux noise can be introduced as

$\Gamma_{\varphi}^{(1)} = {A{{\frac{\partial\omega_{01}}{\partial\Phi_{{ext},2}}}.}}$

In this example, this expression of the dephasing rate is valid when theflux is sufficiently small. If the derivative of the qubit frequency iszero at a flux value, the flux value may be considered a flux sweet spotand the coherence time (T₂) may become indefinitely long (e.g.,virtually infinite). In some cases, the second order contributionneglected in Eq. (3) may dominate at the flux sweet spot. In that case,the contribution of the second order term to the dephasing rate can beestimated through the formula

$\Gamma_{\varphi}^{(2)} \approx {\frac{1}{A^{2}\pi^{2}}\frac{\partial^{2}\omega_{01}}{\partial\Phi_{{ext},2}^{2}}}$

In some cases, the total dephasing rate due to flux noise can beexpressed as the sum of all order derivatives

$\Gamma_{\varphi} = {\sum\limits_{i}\; \Gamma_{\Phi}^{(i)}}$

and is related to T₂ as

${\frac{1}{T_{2}} = {\frac{1}{2\; T_{1}} + \Gamma_{\varphi} + {{other}\mspace{14mu} {dephasing}\mspace{14mu} {rates}}}},$

where T₁ is the energy relaxation time for the qubit.

In some cases (e.g., in the examples shown in FIGS. 4, 5A, and 5B), oneor more flux sweet spots reside at the minima and maxima of the qubitfrequency, and one or more additional flux sweet spots reside betweenthe minima and maxima of the qubit frequency. For instance, the qubitfrequency as a function of the magnetic flux can include a minimum, amaximum, and one or more flux sweet spots between the minimum andmaximum. In some instances, the quantum state of the qubit device 200has reduced sensitivity to magnetic flux noise when the qubit device 200is operated with the qubit frequency tuned to a flux sweet spot. Forexample, the first derivative of the qubit frequency as a function ofthe external flux can be zero at a flux sweet spot. In some cases, whenthe qubit device 200 is operated at a flux sweet spot, second derivativemay dominate the rate of decay of the off-diagonal terms in the qubitdensity matrix; the first-order terms may approach zero. When the firstorder terms approach zero, the characteristic dephasing time (T₂) mayapproach infinity. Accordingly, operating the qubit device 200 at a fluxsweet spot can extend the coherence time and provide better performanceof the qubit device.

In some implementations of the qubit device 200 shown in FIG. 2A, aratio of the first Josephson energy (E_(J) ₁ ) to the second Josephsonenergy (E_(J) ₂ ) can be equal to an integer value (e.g., exactly,approximately or substantially an integer). For example, the values ofthe Josephson energies can be selected such that

$r_{JE} = {\frac{E_{J_{1}}}{E_{J_{2}}} = N_{1}}$

where r_(JE) is the Josephson energy ratio, and N₁ is a non-zero integervalue (e.g., 1, 2, 3, 4, or a higher integer value). In some cases, theJosephson energy ratio is substantially an integer value, for example,within ±0.10 of an integer value, or as close to an integer value as canbe achieved by the selected fabrication processes. In some cases, theJosephson energy ratio can be a real value that is not an integer.

In some implementations of the qubit device 200 shown in FIG. 2A, valuesof the magnetic flux areas (A₁, A₂) can be selected and the externalmagnetic field can be configured such that ratio of the externalmagnetic fluxes (Φ_(ext,1), Φ_(ext,2)) is equal to an integer value(e.g., exactly, approximately or substantially an integer). For example,the magnetic flux parameters can be selected such that

$r_{\Phi} = {\frac{\Phi_{{ext},1}}{\Phi_{{ext},2}} = N_{2}}$

where r_(Φ) is the magnetic flux ratio, and N₂ is a non-zero integervalue (e.g., 1, 2, 3, 4, or a higher integer value). In some cases, themagnetic flux ratio is substantially an integer value, for example,within ±0.10 of an integer value, or as close to an integer value as canbe achieved by the selected fabrication processes. In some cases, themagnetic flux ratio can be a real value that is not an integer.

In some implementations, the qubit device 200 can be configured suchthat the Josephson energy ratio (r_(JE)) of the qubit device equals(e.g., exactly, approximately or substantially equals) a magnetic fluxratio (r_(Φ)) of the qubit device 200 minus one (i.e., r_(JE)=r_(Φ)−1).In some cases, a qubit device can be configured such that a Josephsonenergy ratio substantially equals a magnetic flux ratio minus one, forexample, within ±0.10 of exact equality, or as close to equality as canbe achieved by the selected fabrication processes. In some cases, thevalues of the Josephson energy ratio and the magnetic flux ratio haveanother relationship.

In some aspects of operation, the qubit device 200 is operated by acontrol system that communicates with a quantum integrated circuit thatincludes the qubit device 200. For example, the control system may tunethe qubit frequency of the qubit device by controlling the level ofcurrent in a flux bias device, thereby controlling the external magneticfluxes (Φ_(ext,1), Φ_(ext,2)). In some instances, during operation ofthe qubit device 200, the external magnetic fluxes (Φ_(ext,1),Φ_(ext,2)) are modified (increased or decreased) to tune the qubitfrequency to a flux sweet spot. In some instances, during operation ofthe qubit device 200, the external magnetic fluxes (Φ_(ext,1),Φ_(ext,2)) are modified (increased or decreased) to tune the qubitfrequency away from a flux sweet spot. Operating the qubit device 200may include, for example, performing readout operations, applyingquantum logic gates or other operations with the qubit frequency beingtuned to a flux sweet spot or away from a flux sweet spot.

FIG. 2B shows an equivalent circuit for another example qubit device250. The example qubit device 250 represented in FIG. 2B has similarcircuit elements as the example qubit device 200 represented in FIG. 2A,with the circuit elements in another arrangement. The example qubitdevice 250 represented in FIG. 2B includes a first Josephson junction(J₁) 252A, a second Josephson junction (J₂) 252B, an inductor 254, and acapacitor 258. In this example, the first Josephson junction (J₁) 252Ahas the first inductance (L_(J) ₁ ) and the first Josephson energy(E_(J) ₁ ); the second Josephson junction (J₂) 252B has the secondinductance (L_(J) ₂ ) and the second Josephson energy (E_(J) ₂ ); theinductor 254 has the inductance L_(A); and the capacitor 258 has thecapacitance C_(J). In the example shown in FIG. 2B, the first and secondJosephson junctions each include a respective parasitic capacitance,which is represented by the capacitance C_(J) of the capacitor 258. Theexample qubit device 250 may include additional or different features,including connections with other circuit devices, in some cases.

In the example qubit device 250 represented in FIG. 2B, the inductor 254is connected between a first circuit node 262A and a second circuit node262B; the first Josephson junction (J₁) 252A is connected in parallelwith the inductor 254 between the first circuit node 262A and the secondcircuit node 262B; the second Josephson junction (J₂) 252B is connectedin parallel with the inductor 254 between the first circuit node 262Aand the second circuit node 262B. As shown, the parasitic capacitanceC_(J) of the Josephson junctions, represented by the capacitor 258, isconnected in parallel with the inductor 254. The circuit elements may beconnected in another manner in some cases.

In some implementations, the example qubit device 250 shown in FIG. 2Bcan be implemented as an superconducting quantum integrated circuit. Theindividual elements of the qubit device 250 shown in FIG. 2B (the firstJosephson junction (J₁) 252A, second Josephson junction (J₂) 252B, theinductor 254, and the capacitor 258) may be implemented and fabricatedas the analogous elements of the qubit device 200 shown in FIG. 2A, orthey may be implemented and fabricated in another manner in some cases.

As shown in FIG. 2B, the example qubit device 250 includes three circuitloops—a first circuit loop 260A, a second circuit loop 260B and a thirdcircuit loop 260C. The first circuit loop 260A includes the firstJosephson junction (J₁) 252A and the inductor 254; the second circuitloop 260B includes the second Josephson junction (J₂) 252B and theinductor 254; and the third circuit loop 260C includes the firstJosephson junction (J₁) 252A and the capacitor 258. The qubit device 250may include additional or different circuit loops in some cases. In someinstances, the qubit device 250 does not include the third circuit loop260C, for example, where the capacitor 258 represents the parasiticcapacitance of the Josephson junctions rather than a separate capacitordevice.

In the example shown in FIG. 2B, the first and second circuit loops260A, 260B are configured to receive, during operation of the qubitdevice 250, a magnetic flux that defines a transition frequency of thequbit device 250. As shown, the first circuit loop 260A has a firstinterior area (A′₁) configured to receive a first portion Φ′_(ext,1) ofthe external magnetic flux; and the second circuit loop 260B has asecond interior area (A′₂) configured to receive a second portionΦ′_(ext,2) of the external magnetic flux. When the magnetic field overthe spatial extent of the magnetic flux areas (A′₁, A′₂) is uniform, theratio of the external magnetic fluxes equals the ratio of the magneticflux areas (r_(Φ′)=Φ′_(ext,2)/Φ′_(ext,1)=A′₂/A′₁ in some instances).

In some implementations, the example qubit device 250 can be configuredto provide a flux sweet spot in the qubit frequency (e.g., thetransition frequency between the ground state and first excited state ofthe qubit), analogous to the flux sweet spots described with respect toFIG. 2A. For example, values of the Josephson energies and the interiorareas can be selected such that the qubit device 250 has multiple fluxsweet spots. In some instances, the qubit device 250 is configured tohave flux sweet spots, for example, by setting Φ_(ext,2)>Φ′_(ext,1),such that the external magnetic flux Φ′_(ext,2) through the secondcircuit loop 260B is larger than the external magnetic flux Φ′_(ext,1)through the first circuit loop 260A.

In some implementations of the qubit device 250 shown in FIG. 2B, aratio of the first Josephson energy (E_(J) ₁ ) to the second Josephsonenergy (E_(J) ₂ ) can be equal to an integer value (e.g., exactly,approximately or substantially an integer). For example, the values ofthe Josephson energies can be selected such that

$r_{JE} = {\frac{E_{J_{1}}}{E_{J_{2}}} = N_{1}}$

where r_(JE) is the Josephson energy ratio, and N₁ is a non-zero integervalue (e.g., 1, 2, 3, 4, or a higher integer value). In someimplementations of the qubit device 250 shown in FIG. 2B, values of themagnetic flux areas (A′₁, A′₂) can be selected and the external magneticfield can be configured such that ratio of the external magnetic fluxes(Φ_(ext,1), Φ_(ext,2)) is equal to an integer value (e.g., exactly,approximately or substantially an integer). For example, the magneticflux parameters can be selected such that

$r_{\Phi^{\prime}} = {\frac{\Phi_{{ext},2}^{\prime}}{\Phi_{{ext},1}^{\prime}} = N_{2}}$

where r_(Φ′) is the magnetic flux ratio, and N₂ is a non-zero integervalue (e.g., 1, 2, 3, 4, or a higher integer value). In someimplementations, the qubit device 250 can be configured such that theJosephson energy ratio (r_(JE)) of the qubit device 250 equals (e.g.,exactly, approximately or substantially equals) the magnetic flux ratio(r_(Φ′)) of the qubit device minus one (i.e., r_(JE)=r_(Φ′)−1). Thequbit device 250 may be implemented with other parameters or anotherconfiguration.

In some aspects of operation, the example qubit device 250 shown in FIG.2B is operated in the same manner, or in a similar manner, as describedwith respect to the qubit device 200 shown in FIG. 2A. For example, acontrol system may tune the qubit frequency of the qubit device 250 bycontrolling the level of current in a flux bias device, therebycontrolling the external magnetic fluxes (Φ_(ext,1), Φ_(ext,2)). In someinstances, during operation of the qubit device 250, the externalmagnetic fluxes (Φ_(ext,1), Φ_(ext,2)) are modified (increased ordecreased) to tune the qubit frequency to a flux sweet spot or away froma flux sweet spot.

FIG. 2C shows an equivalent circuit for an example quantum integratedcircuit 280. The example quantum integrated circuit 280 can be included,for example, in the quantum circuit system 102 shown in FIG. 1 or inanother type of quantum processor system. As shown in FIG. 2C, theexample quantum integrated circuit 280 includes a qubit device 200′,capacitors 283 and 285, a readout resonator 284 and a quantum limitedamplifier 286. A quantum integrated circuit may include additional ordifferent features. In some cases, the quantum integrated circuit 280 isa portion of a larger quantum integrated circuit that includes several(e.g., tens, hundreds, thousands) of qubit devices, readout resonators,quantum amplifiers and other circuit elements, for instance, in alattice structure or another type of arrangement.

The example quantum integrated circuit 280 may be fabricated on asubstrate (e.g., a sapphire substrate, a silicon substrate, etc.) thatsupports one or more layers of superconducting material (e.g., aluminum,niobium, etc.) and insulating materials (e.g., aluminum oxide). Forexample, Josephson junctions may be formed by a layer of aluminum oxidesandwiched between layers of aluminum superconducting material,connections between devices may be formed by superconducting wire (e.g.,aluminum), etc.

As shown in FIG. 2C, the qubit device 200′ is capacitively coupled tothe readout resonator 284 by the capacitors 283, and the quantum limitedamplifier 286 is capacitively coupled to the readout resonator 284 bythe capacitors 285. The qubit device 200′ is communicably coupled to amicrowave source 282, and the quantum limited amplifier 286 iscommunicably coupled to a microwave source 287. In some implementations,the microwave sources 282, 287 are included in a control system (e.g.,the control system 110 shown in FIG. 1 or another type of controlsystem) that is external to the quantum integrated circuit 280. Thecontrol system may be communicably coupled to the quantum integratedcircuit 280, for example, by a signal delivery system (e.g., the signaldelivery system 106 shown in FIG. 1 or another type of signal deliverysystem), connectors (e.g., waveguides, coaxial cables, etc.) or othertypes of connection systems or devices.

The example qubit device 200′ in FIG. 2C has the circuit topology of theexample qubit device 200 shown in FIG. 2A. In some cases, the qubitdevice 200′ in FIG. 2C may be implemented in another manner, forexample, according to the circuit topology of the example qubit device250 shown in FIG. 2B or otherwise. A flux bias device (not shown) may beinductively coupled to the qubit device 200′ to provide a magnetic fluxthat tunes a resonance frequency of the qubit device 200′. The flux biasdevice may be included in the quantum integrated circuit 280 (e.g.,on-chip with the circuit elements shown in FIG. 2C) or the flux biasdevice may be external to the quantum integrated circuit 280.

In some aspects of operation, the microwave source 282 sends microwavesignals (e.g., control signals) to the qubit device 200′, for example,to manipulate the quantum state of the qubit. For instance, themicrowave source 282 may generate a microwave signal at the qubitfrequency, and the microwave signal may be delivered to the qubit device200′ to drive a transition between the computational basis states of thequbit. The microwave signals may be configured to perform a single-qubitgate, a two-qubit gate, or another type of quantum computing operation.

In some aspects of operation, the microwave source 287 sends microwavesignals to the readout resonator 284, for example, to detect the quantumstate of the qubit device 200′. For instance, the readout resonator 284can be probed by a microwave signal (e.g., a readout interrogationsignal) that is generated by the microwave source 287 and delivered tothe readout resonator 284 through the quantum limited amplifier 286 andthe capacitors 285; and in response, the readout resonator 284 canproduce a microwave signal (e.g., a readout response signal). Thereadout response signal can be produced by reflecting the readoutinterrogation signal with additional information. In some instances, theresonance frequency of the readout resonator 284 is influenced by thequantum state of the qubit device 200′, and the properties of thereadout response signal indicate a quantum state of the qubit device200′. The additional information can be, for example, a frequency shift,a phase shift, an amplitude shift, or a combination of these and othermodifications, that indicates the state of the qubit device 200′.

In some implementations, the quantum integrated circuit 280 includesadditional qubit devices; the additional qubit devices may be the sameas or similar to the example qubit devices 200, 250 shown in FIGS. 2A,2B, or the additional qubit devices may include different types of qubitdevices (e.g., flatsonium, transmon, fluxonium, or another type of qubitdevice). A flatsonium qubit device is a flux qubit device of the typedescribed with respect to FIGS. 2A, 2B and 3B. A transmon qubit deviceis a charge qubit device that typically includes a single Josephsonjunction and a shunt capacitance. A fluxonium qubit device is a fluxqubit device that typically includes a single Josephson junction, ashunt capacitance and a shunt inductance.

In some cases, the quantum integrated circuit 280 includes another qubitdevice (e.g., on the same substrate) that can be coupled to the examplequbit device 200′ shown in FIG. 2C, for example, to perform a two-qubitlogic gate or another type of quantum logic operation. The other qubitdevice can be, for example, a flatsonium qubit, a transmon qubit, afluxonium qubit or another type of qubit. In some instances, a two-qubitgate may be applied to the two qubits, for example, by tuning the qubitdevice 200′ to a qubit frequency that is equal to (or near) the otherqubit device's qubit frequency.

In some examples, a controlled two-qubit gate operation can be realizedby tuning the qubit frequency of one qubit device towards the qubitfrequency of the second qubit device. For instance, a SWAP gate betweentwo qubits may be applied by first parking the first qubit at afrequency far (e.g., 1 GHz or more) from the second qubit's frequency,slowly tuning the first qubit to the second qubit's frequency, waitingfor some time, and tuning the first qubit back to the parking frequency.For example, in some cases, an iSWAP gate (00→00, 01→i10, 10→i01, 11→11)can be realized by waiting for a time period t=π/2g, where g is thecoupling strength between the two qubits. Other classes of SWAP gatescan be implemented by varying the waiting time. In some implementations,other types of controlled quantum logic gates can be implemented bytuning the frequency of the quit device.

FIGS. 3A and 3B show an example quantum integrated circuit 300. FIG. 3Ashows a top view of a chip that includes a qubit device 306 and anon-chip flux bias device 308; FIG. 3B shows a zoomed-in view of thequbit device 306 and a portion of the nearby flux bias device 308. Theexample quantum integrated circuit 300 can be included, for example, inthe quantum circuit system 102 shown in FIG. 1 or in another type ofquantum processor system.

As shown in FIGS. 3A and 3B, the example quantum integrated circuit 300is implemented on a chip that includes a substrate 301, signal ports 302and 312, a readout resonator 304, the qubit device 306, the flux biasdevice 308 and a filter 310. The chip may include additional ordifferent components or features. In some cases, the quantum integratedcircuit 300 is implemented on a chip that includes additional circuitelements, for instance, an array of qubit devices and associated readoutresonators and filters.

The example substrate 301 can be silicon, sapphire or another type ofmaterial. In some cases, a wafer is processed (e.g., by deposition,lithography, etching, and other fabrication processes), and thesubstrate 301 is cut from the processed wafer to form the chip structureshown in FIG. 3A. The devices on the substrate 301 can be formed by oneor more layers of superconducting material (e.g., aluminum, niobium,etc.) and insulating materials (e.g., aluminum oxide). For example,Josephson junctions in the qubit device 306 may be formed by a layer ofaluminum oxide sandwiched between layers of aluminum superconductingmaterial; drive lines and other connectors, inductors and capacitor padsmay be formed by superconducting material (e.g., aluminum) on thesubstrate 301; etc.

The example readout resonator 304 can be implemented as a lumped-elementelectromagnetic device having an LC characteristic that defines aresonance frequency. For instance, the readout resonator 304 may have acapacitance and inductance that define a resonance characteristic andthat allow the readout resonator 304 to interact with the qubit device306, for example, to detect the state of the qubit device 306. Theexample readout resonator 304 shown in FIG. 3A is a microwave resonator,having a microwave resonance frequency. The readout resonator 304 mayinclude an inductance formed by a meander inductor, a spiral inductor,or an inductor having another shape integrated into a coplanar waveguidetopology or microstrip topology. The readout resonator 304 may include acapacitance formed by one or more capacitive pads that at leastpartially surround the inductor.

The example signal port 302 can be connected to an another device orsystem, for example, to another device on the chip or to a systemexternal to the chip. As shown in FIG. 3A, the signal port 302 isconnected to a resonator drive line 303 that extends from the signalport 302 to a terminal end that is capacitively coupled to the readoutresonator 304. Also shown in FIG. 3A, the readout resonator 304 isconnected to the qubit device 306 by a qubit drive line 305 that extendsbetween the readout resonator 304 and the qubit device 306. In theexample shown, the signal port 302 and the resonator drive line 303 areconfigured to transfer microwave signals between the readout resonator304 and an external control system, for example, through one or moreintermediate systems or components; the signal port 302, the readoutresonator 304, the resonator drive line 303 and the qubit drive line 305are configured to transfer microwave signals between the qubit device306 and the external control system, for example, through one or moreintermediate systems or components.

In some instances, the signal port 302 receives a readout interrogationsignal having a microwave frequency at or near the resonance frequencyof the readout resonator 304. The resonator drive line 303 can transferthe readout interrogation signal to the readout resonator 304, and thereadout resonator 304 can produce a readout response signal byreflecting the readout interrogation signal. The phase of the readoutresponse signal can indicate the quantum state of qubit device 306. Forexample, the quantum state of the qubit device 306 may influence theimpedance of the readout resonator 304 at the frequency of the readoutinterrogation signal.

In some instances, the signal port 302 receives a qubit drive signalhaving a microwave frequency at or near the qubit frequency of the qubitdevice 306. The resonator drive line 303, the readout resonator 304 andthe qubit drive line 305 can transfer the qubit drive signal to thequbit device 306, and the qubit drive signal can drive the qubit device306 at the qubit frequency. For instance, the qubit frequency may beshifted from (e.g., below or above) the resonance frequency of thereadout resonator 304, and the readout resonator 304 may therefore passthe qubit drive signal from the resonator drive line 303 to the qubitdrive line 305. The qubit drive signal can be configured to perform asingle-qubit gate or another type of quantum logic operation on thequbit. For example, the quantum state of the qubit device 306 may bemodified by the qubit drive signal.

The example filter 310 can be implemented as a filter device thatsuppresses or attenuates electromagnetic signals at or around the qubitfrequency. For instance, the filter 310 may have an LC characteristicand that transfers low-frequency (e.g., direct current) signals betweenthe signal port 312 and the flux bias device 308, for example, tocontrol a magnetic field generated by the flux bias device 308. In somecases, the filter 310 can be implemented as a low-pass filter, aband-pass filter, or another type of filter. The filter 310 can beconfigured to suppress information leakage or other types of loss fromthe qubit device 306. The filter 310 may include an inductance formed bya meander inductor, a spiral inductor, or an inductor having anothershape integrated into a coplanar waveguide topology or microstriptopology. The filter 310 may include a capacitance between the inductorand a ground plane.

The example signal port 312 can be connected to an another device orsystem, for example, another device on the chip or a system external tothe chip. As shown in FIG. 3B, the signal port 312 is connected to aflux bias drive line 307 that extends from the signal port 312 andpasses through the filter 310 to the flux bias device 308. In someinstances, the signal port 312 and flux bias drive line 307 transferdirect-current signals between the flux bias device 308 and an externalcontrol system, for example, through one or more intermediate systems orcomponents.

As shown in FIG. 3B, the example qubit device 306 includes a firstJosephson junction 352A, a second Josephson junction 352B, an inductor354 and connection lines 371A, 371B, 371C, 371D. The example qubitdevice 306 in FIG. 3B is an example implementation of the example qubitdevice 250 represented by the equivalent circuit in FIG. 2B. Forexample, the first Josephson junction 352A in FIG. 3B has the firstinductance (L_(J) ₁ ) and the first Josephson energy (E_(J) ₁ ); thesecond Josephson junction 352B in FIG. 3B has the second inductance(L_(J) ₂ ) and the second Josephson energy (E_(J) ₂ ); the inductor 354in FIG. 3B has the inductance L_(A); and the Josephson junctions 352A,352B have the parasitic capacitance C_(J) shunted by the linearinductance (L_(A)) of the inductor 354.

In the example qubit device 306 represented in FIG. 3B, the inductor 354is connected between two circuit nodes on either side of the secondJosephson junction 352B; the first Josephson junction 352A is connected(via the connection lines 371A, 371B, 371C, 371D) in parallel with theinductor 354 between the two circuit nodes; and the second Josephsonjunction 352B is connected in parallel with the inductor 354 between thetwo circuit nodes. The circuit elements may be connected in anothermanner in some cases.

As shown in FIG. 3B, the example qubit device 306 includes two circuitloops—a first circuit loop that includes the first Josephson junction352A and the inductor 354, and a second circuit loop that includes thesecond Josephson junction 352B and the inductor 354. Each of the circuitloops in qubit device 306 defines an interior area that can receivemagnetic flux during operation of the qubit device 306. The firstcircuit loop defines a first magnetic flux area 360A at the boundariesof the connection lines 371A, 371B, 371C, 371D, the first Josephsonjunction 352A and the inductor 354. In particular, the perimeter of thefirst magnetic flux area 360A follows an outer boundary of the inductor354 and an inner boundary of the connection lines 371A, 371B, 371C, 371Dand the first Josephson junction 352A. The second circuit loop defines asecond magnetic flux area 360B at the boundaries of the second Josephsonjunction 352B and the inductor 354. In particular, the perimeter of thesecond magnetic flux area 360B follows an inner boundary of the inductor354 and an inner boundary of the second Josephson junction 352B. Thequbit device 306 may include additional or different circuit loops, andthe circuit loops may be configured in another manner in some cases.

As shown in FIG. 3B, the first magnetic flux area 360A and the secondmagnetic flux area 360B are exclusive areas on the substrate 301. Forinstance, the first magnetic flux area 360A does not contain any of thesecond magnetic flux area 360B, and vice versa. Although the firstmagnetic flux area 360A and the second magnetic flux area 360B share aboundary (the inductor 354), the first magnetic flux area 360A residesentirely outside the second magnetic flux area 360B in the exampleshown.

The magnetic flux received by the first and second magnetic flux areas360A, 360B can define transition frequencies (including the qubitfrequency) of the qubit device 306. In particular, the first magneticflux area 360A can be the first interior area (A′₁, shown in FIG. 2B)that is configured to receive a first portion Φ′_(ext,1) of the externalmagnetic flux; and the second magnetic flux area 360B can be the secondinterior area (A′₂, shown in FIG. 2B) configured to receive a secondportion Φ′_(ext,2) of the external magnetic flux. In some cases, thequbit device 306 shown in FIG. 3B may be configured and operated asdescribed with respect to the qubit devices 200, 250 shown in FIGS. 2A,2B respectively. For instance, the Josephson energy ratio (r_(JE)) ofthe qubit device 306 and the magnetic flux ratio (r_(Φ′)) of the qubitdevice 306 can have the values or the relationships (or both) asdescribed with respect to FIG. 2B.

The example flux bias device 308 can be implemented as a coil that canproduce a magnetic field in the first and second magnetic flux areas360A, 360B. Accordingly, the magnetic field generated by the flux biasdevice 308 contributes to the magnetic flux in the first and secondmagnetic flux areas 360A, 360B during operation. In some instances, themagnetic field produced by the flux bias device 308 varies over thespatial extent of the qubit device 306. In some cases, the flux biasdevice 308 receives a direct current (or other low-frequency) biassignal that causes the flux bias device to generate a constant (orslowly varying) magnetic field. The DC bias signal can be changed, forinstance, to control the strength of the external magnetic fieldexperienced by the qubit device 306. For instance, the magnetic fluxreceived by the first and second magnetic flux areas 360A, 360B can beset at least partially by the DC bias signal.

FIG. 4 is a plot 400 showing transition frequencies as a function ofexternal magnetic flux for an example qubit device. To compute thetransition frequencies shown in FIG. 4, the example qubit device 200represented by the equivalent circuit in FIG. 2A was numerically modeledon a classical computer. The classical Lagrangian for the modeled qubitdevice can be expressed

$\mathcal{L} = {{\frac{1}{2}C_{J}{\overset{.}{\Phi}}^{2}} + {E_{J_{1}}{\cos \left( \frac{2{\pi\Phi}}{\Phi_{0}} \right)}} + {E_{J_{2}}{\cos \left\lbrack {\frac{2\pi}{\Phi_{0}}\left( {\Phi + \Phi_{{ext},1}} \right)} \right\rbrack}} - {\frac{1}{2}{E_{L}\left\lbrack {\frac{2\pi}{\Phi_{0}}\left( {\Phi + \Phi_{{ext},2}} \right)} \right\rbrack}^{2}}}$

Here, Φ represents the node flux at the first circuit node 212A in theexample qubit device represented in FIG. 2A; Φ₀ represents a fluxquantum. Introducing a new variable,

${\phi = {\frac{2\pi}{\Phi_{0}}\left( {\Phi + \Phi_{{ext},2}} \right)}},$

the corresponding Hamiltonian has the form

$H = {\frac{Q^{2}}{2\; C_{J}} - {E_{J_{1}}{\cos \left( {\phi - \frac{2{\pi\Phi}_{{ext},2}}{\Phi_{0}}} \right)}} - {E_{J_{2}}{\cos \left( {\phi - {2\pi \frac{\Phi_{{ext},2} - \Phi_{{ext},1}}{\Phi_{0}}}} \right)}} - {\frac{1}{2}E_{L}{\phi^{2}.}}}$

Expressing the total charge as the net number of Cooper pairs thattunneled through the junctions, Q=2en (where e represents the electroncharge) we have

$\frac{{\hat{Q}}^{2}}{2\; C_{J}} = {4\; E_{C}{{\hat{n}}^{2}.}}$

Noting that the number of Cooper pairs {circumflex over (n)} and thephase {circumflex over (φ)} are conjugate variables, we can write thequantized Hamiltonian as

$H = {{4\; E_{C}{\hat{n}}^{2}} - {E_{J_{1}}{\cos \left( {\hat{\phi} - \frac{2{\pi\Phi}_{{ext},2}}{\Phi_{0}}} \right)}} - {E_{J_{2}}{\cos \left( {\hat{\phi} - {2\pi \frac{\Phi_{{ext},2} - \Phi_{{ext},1}}{\Phi_{0}}}} \right)}} + {\frac{1}{2}E_{L}{{\hat{\phi}}^{2}.}}}$

To generate the plot 400 shown in FIG. 4, the qubit device was modeledwith a global magnetic field applied, where the fluxes Φ_(ext,1) andΦ_(ext,2) are related to their respective loop areas. In this example,the magnetic fluxes are related by the magnetic flux ratio asΦ_(ext,1)=r_(Φ)Φ_(ext,2). In view of this, the Hamiltonian now takes theform

$H = {{4\; E_{C}{\hat{n}}^{2}} - {E_{J_{1}}{\cos \left( {\hat{\phi} - \frac{2{\pi\Phi}_{{ext},2}}{\Phi_{0}}} \right)}} - {E_{J_{2}}{\cos \left( {\hat{\phi} - {\left( {1 - r_{\Phi}} \right)\frac{2{\pi\Phi}_{{ext},2}}{\Phi_{0}}}} \right)}} + {\frac{1}{2}E_{L}{{\hat{\phi}}^{2}.}}}$

In the numerical model used to generate the plot 400, the followingexample values were chosen for the circuit parameters:

${\frac{E_{J_{1}}}{E_{J_{2}}} = 3},{E_{J_{1}} = {{\frac{1}{L_{J_{1}}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {3\mspace{14mu} {GHz}}}},{E_{J_{2}} = {{\frac{1}{L_{J_{2}}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {9\mspace{14mu} {GHz}}}},{E_{L} = {{\frac{1}{L_{A}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {0.45\mspace{14mu} {GHz}}}},{E_{C} = {\frac{e^{2}}{2\; C_{J}} = {3\mspace{14mu} {GHz}}}},{r_{\Phi} = 4.}$

Here, E_(L) represents the energy of the inductor 204, and E_(C)represents the charging energy of the capacitor 208. The Hamiltonianshown above can be numerically diagonalized to compute the energyspectrum shown in FIG. 4.

The plot 400 includes a horizontal axis 402 representing a range ofvalues for the external magnetic flux Φ_(ext,2) in units of flux quantaΦ₀. The plot 400 includes a vertical axis 404 representing a range ofvalues for the plotted transition frequencies f₀₁, f₀₂, f₀₃ in units ofgigahertz (GHz). A first curve 406A in the plot 400 represents thetransition frequency f₀₁, which is the frequency associated with thetransition between the ground state and first excited state of the qubitdevice; a second curve 406B in the plot 400 represents the transitionfrequency f₀₂, which is the frequency associated with the transitionbetween the ground state and second excited state of the qubit device;and a third curve 406C in the plot 400 represents the transitionfrequency f₀₃, which is the frequency associated with the transitionbetween the ground state and third excited state of the qubit device.The large gap between the first curve 406A and the second curve 406B inthe plot 400 (between the first and second transition frequencies f₀₁,f₀₂) shows that the qubit device is strongly anharmonic.

The qubit device may be operated with the ground state and first excitedstates representing the computational basis states of a qubit. In suchcases, the first transition frequency f₀₁ (shown as the first curve406A) represents the qubit frequency. As shown in FIG. 4, the qubitfrequency has multiple flux sweet spots. The example flux sweet spots408A, 408B, 408C, 408D labeled in FIG. 4 are locations where the firstcurve 406A has an inflection point (first derivative equal to zero)between the local maxima and minima of the first curve 406A. Inparticular, in the range of external flux shown in FIG. 4 (minus one toplus one flux quantum), the first curve 406A has a maximum value near7.5 GHz at three locations (at Φ_(ext,2)/Φ₀={−1, 0, 1}); the first curve406A has a minimum value near 0.1 GHz at two locations (atΦ_(ext,2)/Φ₀={−0.5, 0.5}); and the first curve 406A has four otherinflection points between the maximum and minimum values. The four otherinflection points indicate flux sweet spots in the qubit frequency near4.0 GHz at four flux points (at Φ_(ext,2)/Φ₀={−0.75, −0.25, 0.25,0.75}). As shown in FIG. 4, the total number of flux sweet spots over asingle flux quantum (e.g., at {0.00, 0.25, 0.50, 0.75} in the fluxquantum from Φ_(ext,2)=[0, Φ₀] in the plot) equals the magnetic fluxratio r₀=4.

When the qubit device is operated with its qubit frequency tuned to oneof the flux sweet spots 408A, 408B, 408C, 408D, the characteristicdephasing time of the qubit device can approach the limits set bydissipation in the system. Moreover, the qubit frequency demonstrates acharacteristic linear magnetic flux dependence away from the flux sweetspots 408A, 408B, 408C, 408D, which can be useful for performing logicgates or applying other operations to the qubit device in some cases.

FIGS. 5A and 5B are plots 500, 550 showing transition frequencies as afunction of external magnetic flux for example qubit devices. Each ofthe curves plotted in FIGS. 5A and 5B represents a qubit device having adistinct value of the magnetic flux ratio r_(Φ). In the examples shown,the maximum frequency for each curve is determined by the inductanceenergy E_(L), while the frequencies at which the flux sweet spots appearare controlled by the asymmetry of the two Josephson junctions.

The plots 500, 550 each include a horizontal axis 502 representing arange of values for the external magnetic flux Φ_(ext,2) in units offlux quanta Φ₀. The plots 500, 550 each include a vertical axis 504representing a range of values for the first transition frequency f₀₁for each qubit device in units of gigahertz (GHz). Here, the firsttransition frequency f₀₁ represents the qubit frequency.

In FIG. 5A, a first curve 506A in the plot 500 represents the qubitfrequency for a qubit device having a magnetic flux ratio r_(Φ)=2, whichhas two flux sweet spots over one flux quantum (e.g., at {0.00, 0.50} inthe flux quantum from Φ_(ext,2)/Φ₀=[0,1] in the plot). A second curve506B in the plot 500 represents the qubit frequency for a qubit devicehaving a magnetic flux ratio r_(Φ)=4, which has four flux sweet spotsover one flux quantum (e.g., at {0.00, 0.25, 0.50, 0.75} in the fluxquantum from Φ_(ext,2)/Φ₀=[0,1] in the plot). A third curve 506C in theplot 500 represents the qubit frequency for a qubit device having amagnetic flux ratio r_(Φ)=6, which has six flux sweet spots over oneflux quantum (e.g., at {0.00, 0.17, 0.33, 0.50, 0.67, 0.83} in the fluxquantum from Φ_(ext,2)/Φ₀=[0,1] in the plot).

In FIG. 5B, a fourth curve 506D in the plot 550 represents the qubitfrequency for a qubit device having a magnetic flux ratio r_(Φ)=3, whichhas four flux sweet spots over one flux quantum (e.g., at {0.00, 0.25,0.50, 0.75} in the flux quantum from Φ_(ext,2)/Φ₀=[0,1] in the plot). Afifth curve 506E in the plot 550 represents the qubit frequency for aqubit device having a magnetic flux ratio r_(Φ)=5, which has six fluxsweet spots over one flux quantum (e.g., at {0.00, 0.17, 0.33, 0.50,0.67, 0.83} in the flux quantum from Φ_(ext,2)/Φ₀=[0,1] in the plot). Asixth curve 506F in the plot 550 represents the qubit frequency for aqubit device having a magnetic flux ratio r_(Φ)=7, which has eight fluxsweet spots over one flux quantum (e.g., at {0.125, 0.25, 0.375, 0.50,0.625, 0.75, 0.875} in the flux quantum from Φ_(ext,2)/Φ₀=[0,1] in theplot).

In the examples shown in FIGS. 5A and 5B, the number of flux sweet spotsis related to the ratio of the magnetic fluxes (r_(Φ)). For instance, inthe examples shown, the number n_(sp) of sweet spots is given by

$n_{sp} = \left\{ \begin{matrix}{r_{\Phi},} & {{{for}\mspace{14mu} {even}\mspace{14mu} r},} \\{{r_{\Phi} + 1},} & {{for}\mspace{14mu} {odd}\mspace{14mu} {r.}}\end{matrix} \right.$

The number of sweet spots may have another value in some cases.

FIG. 6 is a plot 600 showing the derivative of a transition frequency asa function of external magnetic flux for an example qubit device. Theplot 600 includes a horizontal axis 602 representing a range of valuesfor the external magnetic flux Φ_(ext,2) in units of flux quanta Φ₀. Theplot 600 includes a vertical axis 604 representing a range of values forthe derivative of the first transition frequency f₀₁ (in units of GHz)with respect to the external magnetic flux Φ_(ext,2) (in units of fluxquanta Φ₀).

To compute the values represented by the curve 606 shown in FIG. 6, theexample qubit device 200 represented by the equivalent circuit in FIG.2A was numerically modeled on a classical computer. In the numericalmodel used to generate the plot 600, the following example values werechosen for the circuit parameters:

${\frac{E_{J_{1}}}{E_{J_{2}}} = 3},{E_{J_{1}} = {{\frac{1}{L_{J_{1}}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {3\mspace{14mu} {GHz}}}},{E_{J_{2}} = {{\frac{1}{L_{J_{2}}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {9\mspace{14mu} {GHz}}}},{E_{L} = {{\frac{1}{L_{A}}\left( \frac{\Phi_{0}}{2\pi} \right)^{2}} = {0.45\mspace{14mu} {GHz}}}},{E_{C} = {\frac{e^{2}}{2\; C_{J}} = {3\mspace{14mu} {GHz}}}},{r_{\Phi} = 4.}$

In the example shown in FIG. 6, the first derivative of the qubitfrequency is zero at the engineered flux sweet spots. As shown, the fluxsweet spots have a finite bandwidth, which allows the qubit frequency tobe parked at a flux sweet spot while other qubit operations (e.g., qubitreadout, single and two qubit gates) are performed.

In a general aspect of the examples described here, a quantum integratedcircuit includes multiple circuit loops.

In a first example, a qubit device includes an inductor connectedbetween a first circuit node and a second circuit node; a firstJosephson junction connected in parallel with the inductor between thefirst circuit node and the second circuit node; and a second Josephsonjunction connected in parallel with the inductor between the firstcircuit node and the second circuit node.

In a second example, a quantum computing method includes operating aqubit device. The qubit device includes an inductor connected between afirst circuit node and a second circuit node; a first Josephson junctionconnected in parallel with the inductor between the first circuit nodeand the second circuit node; and a second Josephson junction connectedin parallel with the inductor between the first circuit node and thesecond circuit node.

Implementations of the first or second example may include one or moreof the following features. The qubit device can include a first circuitloop that includes the first Josephson junction; and a second circuitloop that includes the second Josephson junction. The first and secondcircuit loops can be configured to receive, during operation of thequbit device, a magnetic flux that defines a transition frequency of thequbit device.

Implementations of the first or second example may include one or moreof the following features. The first circuit loop can include the firstJosephson junction and the second Josephson junction; and the secondcircuit loop can include the inductor and the second Josephson junction.The first circuit loop can include the first Josephson junction and theinductor; and the second circuit loop can include the second Josephsonjunction and the inductor.

Implementations of the first or second example may include one or moreof the following features. The first Josephson junction can have a firstJosephson energy; the second Josephson junction can have a secondJosephson energy. The first and second Josephson energies can be unequal(e.g., substantially different values). A ratio of the first Josephsonenergy to the second Josephson energy can be substantially an integervalue (e.g., within ±0.10 of an integer value). The first circuit loopcan define a first magnetic flux area to receive a first portion of themagnetic flux; the second circuit loop can define a second magnetic fluxarea to receive a second portion of the magnetic flux. The first andsecond portions of the magnetic flux can be unequal (e.g., substantiallydifferent values). A ratio of the first portion of the magnetic flux tothe second portion of the magnetic flux can be substantially an integervalue (e.g., within ±0.10 of an integer value).

Implementations of the first or second example may include one or moreof the following features. The first Josephson junction can have a firstJosephson energy; the second Josephson junction can have a secondJosephson energy; the first circuit loop can define a first magneticflux area to receive a first portion of the magnetic flux; the secondcircuit loop can define a second magnetic flux area to receive a secondportion of the magnetic flux; and the qubit device can be configuredsuch that a Josephson energy ratio of the qubit device substantiallyequals (e.g., within ±0.10) a magnetic flux ratio of the qubit deviceminus one. The magnetic flux ratio can be the ratio of the first portionof the magnetic flux to the second portion of the magnetic flux, and theJosephson energy ratio can be the ratio of the first Josephson energy tothe second Josephson energy. The magnetic flux ratio can be the ratio ofthe second portion of the magnetic flux to the first portion of themagnetic flux, and the Josephson energy ratio can be the ratio of thefirst Josephson energy to the second Josephson energy.

Implementations of the first or second example may include one or moreof the following features. The transition frequency as a function of themagnetic flux can include a minimum, a maximum, and a flux sweet spotbetween the minimum and maximum.

Implementations of the first or second example may include one or moreof the following features. The first Josephson junction and the secondJosephson junction can each include a respective parasitic capacitancein parallel with the inductor. The inductor can be a superinductor thatincludes a serial array of Josephson junctions. The inductor can be asuperinductor that includes a superconducting wire.

Implementations of the first or second example may include one or moreof the following features. Operating the qubit device can includecontrolling a magnetic flux received by first and second circuit loopsof the qubit device, where the first circuit loop includes the firstJosephson junction, and the second circuit loop includes the secondJosephson junction. Controlling the magnetic flux can tune a transitionfrequency of the qubit device.

Implementations of the first or second example may include one or moreof the following features. Controlling the magnetic flux can includetuning the transition frequency of the qubit device to a flux sweetspot. Operating the qubit device can include performing a readout of thequbit device while the transition frequency is tuned to the flux sweetspot. Controlling the magnetic flux can include tuning the transitionfrequency away from a flux sweet spot. Operating the qubit device caninclude applying a logic gate to the qubit device while the transitionfrequency is tuned away from the flux sweet spot.

In a third example, a quantum integrated circuit includes a firstcircuit loop and a second circuit loop. The first circuit loop defines afirst magnetic flux area configured to receive a first magnetic fluxduring operation of the quantum integrated circuit. The first circuitloop includes a first Josephson junction having a first Josephsonenergy. The second circuit loop defines a second magnetic flux areaconfigured to receive a second magnetic flux during operation of thequantum integrated circuit. The second circuit loop includes a secondJosephson junction having a second Josephson energy. The secondJosephson junction is connected in parallel with the first Josephsonjunction. Either or both of the first circuit loop and the secondcircuit loop include an additional circuit element that is connected inparallel with the first Josephson junction and the second Josephsonjunction.

Implementations of the third example may include one or more of thefollowing features. A Josephson energy ratio of the first and secondcircuit loops can substantially equal a magnetic flux ratio of the firstand second circuit loops minus one. The Josephson energy ratio can bethe ratio of the first Josephson energy to the second Josephson energy,and the magnetic flux ratio can be either the ratio of the firstmagnetic flux area to the second magnetic flux area or the ratio of thesecond magnetic flux area to the first magnetic flux area.

Implementations of the third example may include one or more of thefollowing features. The magnetic flux received by the first and secondcircuit loops can define a transition frequency between energy states inthe quantum integrated circuit.

Implementations of the third example may include one or more of thefollowing features. The first circuit loop can include the firstJosephson junction and the second Josephson junction; and the secondcircuit loop can include the additional circuit element and the secondJosephson junction. The first circuit loop can include the firstJosephson junction and the additional circuit element; and the secondcircuit loop can include the second Josephson junction and theadditional circuit element.

Implementations of the third example may include one or more of thefollowing features. A ratio of the first Josephson energy to the secondJosephson energy can be substantially an integer value. A ratio of thefirst magnetic flux to the second magnetic flux can be substantially aninteger value.

Implementations of the third example may include one or more of thefollowing features. The quantum integrated circuit can include a qubitdevice. The qubit device can include the first Josephson junction, thesecond Josephson junction and the additional circuit element, and theadditional circuit element can include an inductor.

Implementations of the third example may include one or more of thefollowing features. The quantum integrated circuit can include amicrowave resonator device capacitively coupled to the qubit device. Themicrowave resonator device can reside on a chip with the qubit device.The quantum integrated circuit can include a flux bias device thatresides on a chip with the qubit device. The flux bias device can beconfigured to generate a magnetic field that contributes to the firstmagnetic flux and the second magnetic flux during operation of thequantum integrated circuit.

In a fourth example, a quantum computing system includes a qubit device.The qubit device includes a first superconducting quantum interferencedevice (SQUID) loop and a second SQUID loop. The first and second SQUIDloops are configured to receive a magnetic flux that defines atransition frequency of the qubit device.

Implementations of the fourth example may include one or more of thefollowing features. The first and second SQUID loops can be configuredsuch that the transition frequency as a function of the magnetic fluxhas a maximum, a minimum and a flux sweet spot between the maximum andthe minimum.

Implementations of the fourth example may include one or more of thefollowing features. The first SQUID loop can include a first Josephsonjunction, the second SQUID loop can include a second Josephson junctionand the first SQUID loop or the second SQUID loop (or both) can includea superinductor. The first SQUID loop can include the first Josephsonjunction and the second Josephson junction, and the second SQUID loopcan include the superinductor and the second Josephson junction. Thefirst SQUID loop can include the first Josephson junction and thesuperinductor, and the second SQUID loop can include the secondJosephson junction and the superinductor.

Implementations of the fourth example may include one or more of thefollowing features. The first Josephson junction can have a firstJosephson energy; the second Josephson junction can have a secondJosephson energy; the first SQUID loop can define a first magnetic fluxarea to receive a first portion of the magnetic flux; and the secondSQUID loop can define a second magnetic flux area to receive a secondportion of the magnetic flux. The qubit device can be configured suchthat a Josephson energy ratio of the qubit device substantially equals amagnetic flux ratio of the qubit device minus one. The magnetic fluxratio can be the ratio of the first portion of the magnetic flux to thesecond portion of the magnetic flux, and the Josephson energy ratio canbe the ratio of the first Josephson energy to the second Josephsonenergy.

Implementations of the fourth example may include one or more of thefollowing features. The quantum computing system can include a chip thatsupports the qubit device and a readout resonator coupled to the qubitdevice. The quantum computing system can include a quantum circuitsystem that houses the qubit device, and a signal delivery systemconnected to the quantum circuit system. The signal delivery system canbe configured to communicate signals between the quantum circuit systemand a control system.

In a fifth example, a system of integrated microwave quantum circuitsincludes multiple tunable circuit elements for the cancellation ofnoise. In some cases, the tunable elements control the frequencyresponse of a circuit. In some cases, the tunable elements control thecoupling strength between devices. The tunable elements may beimplemented, for example, as Superconducting Quantum InterferenceDevices (SQUID) loops. In some cases, the SQUID loop areas are relatedas integer multiples. In some cases, the Josephson energies are relatedas integer multiples. The SQUID loops can have shared degrees offreedom, for example, a common applied magnetic field. The flux throughthe SQUID loops can be independently controlled. An operating bias canachieve a response function to flux that is approximately zero.

In a sixth example, a quantum integrated circuit includes asuperinductance, a first superconducting loop, and a secondsuperconducting loop, where the second loop is in a branch of the firstloop. In some cases, the area of the first loop is an integer multipleof the area of the second loop. In some cases, the superinductance isdisposed in a branch of the first loop. The first loop, the second loopor both may include a plurality of Josephson junctions. Thesuperinductance may be formed by a serial array of Josephson junctions,a high-kinetic inductance superconducting wire or another type ofstructure. The respective Josephson energies of the Josephson junctionscan be related as integer multiples. In some cases, multiple of suchquantum integrated circuits are implemented with multiplesuperconducting microwave resonators on a single chip.

In a seventh example, a quantum integrated circuit includes circuitloops and a superinductance that can be operated at a flux bias pointwhere a transition frequency of the circuit is insensitive to magneticflux noise. In some instances, a two-qubit gate is performed by applyingan external magnetic field through the circuit loops to tune anoperating frequency away from the magnetic flux insensitivity point. Insome instances, qubit readout is performed at a flux bias point wherethe device is insensitive to magnetic flux noise.

In an eighth example, a quantum computing device includes a qubit devicethat includes an asymmetric SQUID and a large inductor. In some cases,the loop areas formed by the large inductor and Josephson junctions arein integer relation to one another. In some cases, the Josephson energyof the two junctions in the asymmetric SQUID are in integer relation toone another. The qubit device can be capacitively coupled to a linearmicrowave resonator. The qubit device can be capacitively coupled to aplurality of other qubit devices.

Implementations of the eighth example may include one or more of thefollowing features. The qubit device can be controlled with a singleon-chip flux bias device. The qubit device can be controlled with aplurality of on-chip flux bias devices. The qubit device can becontrolled with an externally applied oscillating electric field. Thequbit device can be measured with an externally applied oscillatingfield. The qubit device can be operated at a flux insensitive point. Thequbit device can be operated at a flux-sensitive point. A two-qubit gateprocedure can be implemented between the qubit device and a anothersuperconducting qubit device.

In some examples, a quantum processor can be implemented based on anarray of flux-insensitive qubit devices, such as, for example, the qubitdevices described above. In some examples, a quantum processor can beimplemented based on an array of flux-insensitive qubit devices andcharge-insensitive qubit devices (e.g., transmon qubit devices). In someexamples, a quantum processor can be implemented based on an array offlux-insensitive qubit devices and readout resonators. In some cases,nearest neighbors are capacitively coupled. In some cases, nearestneighbors are inductively coupled.

While this specification contains many details, these should not beconstrued as limitations on the scope of what may be claimed, but ratheras descriptions of features specific to particular examples. Certainfeatures that are described in this specification in the context ofseparate implementations can also be combined. Conversely, variousfeatures that are described in the context of a single implementationcan also be implemented in multiple embodiments separately or in anysuitable subcombination.

A number of embodiments have been described. Nevertheless, it will beunderstood that various modifications can be made. Accordingly, otherembodiments are within the scope of the following claims.

1. A qubit device comprising: an inductor connected between a firstcircuit node and a second circuit node; a first Josephson junctionconnected in parallel with the inductor between the first circuit nodeand the second circuit node; and a second Josephson junction connectedin parallel with the inductor between the first circuit node and thesecond circuit node.
 2. The qubit device of claim 1, comprising: a firstcircuit loop that includes the first Josephson junction; and a secondcircuit loop that includes the second Josephson junction, wherein thefirst and second circuit loops are configured to receive, duringoperation of the qubit device, a magnetic flux that controls a qubitfrequency of the qubit device.
 3. The qubit device of claim 2, wherein:the first circuit loop includes the first Josephson junction and thesecond Josephson junction; and the second circuit loop includes theinductor and the second Josephson junction.
 4. The qubit device of claim2, wherein: the first circuit loop includes the first Josephson junctionand the inductor; and the second circuit loop includes the secondJosephson junction and the inductor.
 5. The qubit device of claim 2,wherein: the first Josephson junction has a first Josephson energy; thesecond Josephson junction has a second Josephson energy; and a ratio ofthe first Josephson energy to the second Josephson energy issubstantially an integer value.
 6. The qubit device of claim 2, wherein:the first Josephson junction has a first Josephson energy; the secondJosephson junction has a second Josephson energy; and the first andsecond Josephson energies are unequal.
 7. The qubit device of claim 2,wherein: the first circuit loop defines a first magnetic flux area toreceive a first portion of the magnetic flux; the second circuit loopdefines a second magnetic flux area to receive a second portion of themagnetic flux; and a ratio of the first portion of the magnetic flux tothe second portion of the magnetic flux is substantially an integervalue.
 8. The qubit device of claim 2, wherein: the first circuit loopdefines a first magnetic flux area to receive a first portion of themagnetic flux; the second circuit loop defines a second magnetic fluxarea to receive a second portion of the magnetic flux; and the firstportion and second portions of the magnetic flux are unequal. 9-10.(canceled)
 11. The qubit device of claim 2, wherein the qubit frequencyas a function of the magnetic flux comprises a minimum, a maximum, and aflux sweet spot between the minimum and maximum.
 12. The qubit device ofclaim 1, wherein the first Josephson junction and the second Josephsonjunction each comprise a respective parasitic capacitance in parallelwith the inductor.
 13. The qubit device of claim 1, wherein the inductorcomprises a superinductor that includes a serial array of Josephsonjunctions.
 14. The qubit device of claim 1, wherein the inductorcomprises a superinductor that includes a superconducting wire. 15-30.(canceled)
 31. A quantum computing method comprising operating a qubitdevice, the qubit device comprising: an inductor connected between afirst circuit node and a second circuit node; a first Josephson junctionconnected in parallel with the inductor between the first circuit nodeand the second circuit node; and a second Josephson junction connectedin parallel with the inductor between the first circuit node and thesecond circuit node. 32-39. (canceled)
 40. A quantum computing systemcomprising: a qubit device comprising a first superconducting quantuminterference device (SQUID) loop and a second SQUID loop, wherein thefirst and second SQUID loops are configured to receive a magnetic fluxthat defines a qubit frequency of the qubit device.
 41. The quantumcomputing system of claim 40, wherein the first and second SQUID loopsare configured such that the qubit frequency as a function of themagnetic flux comprises a maximum, a minimum and a flux sweet spotbetween the maximum and the minimum.
 42. The quantum computing system ofclaim 40, comprising a chip that includes the qubit device and a readoutresonator coupled to the qubit device.
 43. The quantum computing systemof claim 40, wherein the first SQUID loop include a first Josephsonjunction, the second SQUID loop includes a second Josephson junction andat least one of the first SQUID loop or the second SQUID loop includes asuperinductor.
 44. The quantum computing system of claim 43, wherein:the first SQUID loop includes the first Josephson junction and thesecond Josephson junction; and the second SQUID loop includes thesuperinductor and the second Josephson junction.
 45. The quantumcomputing system of claim 43, wherein: the first SQUID loop includes thefirst Josephson junction and the superinductor; and the second SQUIDloop includes the second Josephson junction and the superinductor. 46.(canceled)
 47. The quantum computing system of claim 40, comprising: aquantum circuit system that includes the qubit device; a signal deliverysystem connected to the quantum circuit system and configured tocommunicate signals between the quantum circuit system and a controlsystem.